UNIVERSITY OF CALIFORNIA IRVINE

DEPARTMENT OF LOGIC AND PHILOSOPHY OF SCIENCE

COLLOQUIUM

"The Empty Set, the Singleton, and the Ordered
Pair"

Abstract: "For the modern set theorist the empty set 0 , the
singleton {a} , and the ordered pair are at the beginning of the systematic,
axiomatic development of set theory, both as a field of
mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building blocks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated
long before the complexities of Power Set, Replacement, and Choice are
broached in the formal elaboration of
the `set of'{} operation. So it is surprising that, while these notions
are unproblematic today, they were once
sources of considerable concern and confusion among leading pioneers of
mathematical logic like Frege,
Russell, Dedekind, and Peano. In the development of modern mathematical
logic out of the turbulence of 19th
Century logic, the emergence of the empty set, the singleton, and the
ordered pair as clear and elementary
set-theoretic concepts serves as a motif that reflects, if not
illuminates, larger and more significant
developments in mathematical logic: the shift from the intensional to the
extensional viewpoint, the development
of type distinctions, the logical vs. the iterative conception of set, and
the emergence of various concepts and
principles as distinctively set-theoretic rather than purely logical. Here
there is a loose analogy with Tarski's
recursive definition of truth for formal languages: The mathematical
interest lie mainly in the procedure of
recursion and the attendant formal semantics in model theory, whereas the
philosophical interest lies mainly in
the basis of the recursion, truth and meaning at the level of basic
predication."

1995

"The Emergence of Descriptive Set Theory." Synthese (1995), 251:241-262. Reprinted in Jaakko Hintikka, ed., From Dedekind to Gödel: Essays on the Development of the Foundations of
Mathematics, pp. 241-262. Synthese Library, 251. Dordrecht: Kluwer, 1995.

1996

"The Mathematical Development of Set Theory from Cantor to Cohen."
Bulletin of Symbolic Logic (1996), 2(1):1-71.

1997

(with Burton Dreben.) "Hilbert and Set theory."
Synthese (January 1997), 110(1):77-125.
This issue is entitled "Symposium
David Hilbert," and is edited by Akihiro Kanamori and A. Tauber.